The Separable Convex Quadratic Knapsack Problem

This paper considers the problem of minimizing a convex, separable quadratic function subject to a knapsack constraint and a box constraint. An algorithm called NAPHEAP is developed for solving this problem. The algorithm solves the Karush-Kuhn-Tucker system using a starting guess to the optimal Lagrange multiplier and updating the guess monotonically in the direction of the solution. The starting guess is either computed using a Newton-type method (variable fixing method, secant method, or Newton’s method) or is supplied by the user. A key innovation in our algorithm is the implementation of a heap data structure for storing the break points of the derivative of the dual function. Given a starting guess, the heap is built with about n comparisons, where n is the problem dimension. Each subsequent iteration amounts to an update of the heap, which can be done in about log 2 n comparisons. Hence, when the starting guess is sufficiently good, the cost of the iteration updates can be a small multiple of log 2 n comparisons. In contrast, Newton-type methods can require O(n) operations per iteration, even when the starting guess is near the solution. Hence, a hybrid algorithm that uses a Newton-type method to generate a starting guess, followed by the heap-based monotone break point searching scheme, can be faster than a Newton-type method by itself. Numerical results are presented.